Discrete Mathematics Assignment-3

a) 
  Modus tollen
  De Morgan’s laws
  Distributive laws
  Identity Laws
 
  De Morgan’s laws
  Domination Laws
b) 
   Modus ponens
   Modus tollen
c) 
 
 De Morgan’s laws
 
 

Step2 is wrong: Step 2 should be Existential instantiation using(1)
Step6 is wrong: Step 6 can't be Universal generalization

    P: a is a multiple of 4
   Q: 3a+1 is odd
a)Direct proof:
 : 
     a is a mutiple of 4
     Suppose: a=4n
    
     is an odd
b)Contradiction:
 : 
     is even ()
    
         
      
       is odd ()
    
c)Contraposition:
 : 
     ()
    
     It's clear that can't be divided by 4 ()
   

   If is even then is even then the proof is done.
  If is odd then is an odd then is an even number

    
   
    for n

Basis step: P(1):
Induction Hypothesis: Assume P(k-1) is true

  
Inductive Step:
     
     
     

    
 
Basis step: P(0): 
Induction Hypothesis: Assume P(k) is true
Inductive Step: